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Pressing COM1/2 and ENTR does not switch between ”preset/frequency modes”. ENTR without inputting a preset/freq switches to the last used channel, and can be used for pogoing to another channel and back easily.
Preset channels and manual frequencies can be switched into in the exact same way (COM1/2, preset/freq, ENTR). There’s no ”mode” to change per se. If you really wish to cycle through the presets with selector switch, you can, but it’s faster to just enter the preset number and press enter.

To both of the questions: No, not possible. As far as I know, not possible on the real jet either, but I might be wrong. Also, you didn't ask but no TACAN indications on the HUD either.
So it's quite a bit different than, for example, the Hornet. I assume this is an Air Force / Navy difference, where the AF just hasn't had the need for it, at least not enough to spend resources to add them.

You can use air-to-ground modes in NAV master mode.

That's a big track. It's always better to do as short as possible a track where you can reproduce the issue. Nevertheless, I watched the start and unless you fixed it later in the track by doing a new INS alignment, I found an issue: You're using the wrong procedure for INS normal alignment.
When using the normal (NORM) alignment, you must enter or verify (with ENTR) the current position within 2 minutes of moving the INS knob to NORM. So the correct process is:
Switch INS knob to NORM. Within 2 minutes: Enter correct LAT LNG and SALT. If they're already correct, as they usually are, just press ENTR on all those three lines, verifying the current values. Wait for about 8 minutes for align to complete. Alignment is complete when status is /10 and RDY is flashing. Switch INS knob to NAV. What I saw in your track was as follows. This is as accurate as I remember as I only watched it once. In any case the problem was clear and you could briefly see the borked alignment status on DED:
Switched INS knob to NORM. Waited for the alignment to go to /10. Verified the coordinates with ENTR. Verifying the coordinates after 2 minutes into the alignment process resets the alignment and it starts from zero. You can see the time and status resetting on DED. Switched INS knob to NAV. Last INS status I saw on the DED was something like 0.1/96 so completely borked. Hope this helps. See DCS F-16C Early Access Guide p. 172 for more information.

I think axis controls are not included in the search. What you're looking for is "RDR CURSOR Switch" (Y and X axis, respectively) in Axis Assign list. For the slew axis:

Depress is under "ENABLE Switch - Depress":

The physical slewable+depressable switch is called the "(RDR) CURSOR/ENABLE switch" for example in the DCS F-16C EA manual.

It's not the speed, it's the steering. With DCS F-16 and LGBs, you need to line up the steering line and Flight Path Marker to release.
In the track with autopilot, it seems to align Target Designator box and the steering line. This takes you off right of the correct line, FPM is to the right of steering line.
Line them up like this, and you're good to go:

I had issues with this. Seems like it was caused by my idle cutoff binding (using WH throttle). 
 
Make sure you are in idle cutoff, or it will feed and you will get hot start, even if throttle handle is all the way back and looking like it’s in cutoff. 

And they did add the OFF state for bindings in 2.8, so even if you don't have special else-off bindings, you can do similar yourself. See this video (I keep linking this to different threads because I really like this addition 😄)
 

In your screenshot time in briefing is 17:30 and time on ded is 14:31 (not 16:30). The 3h difference between DED and briefing is because of time zones.
DED time defaults to UTC. Syria is UTC+3 so local time is 3h ahead. Mission Editor and briefing show local times.

Checked the last one. I think you need to Cursor Zero (CZ).
When you slew, the slew is kept in the system as a cursor delta. It is kept and transferred when you switch steerpoints. So if you have slewed, and you switch to another STPT, pod will look at that STPT offset by the delta (slew) that exists in the system. This is often not what you want, so you'll cursor zero (CZ) when you have slews and want to look at another STPT. CZ removes the delta so now it will look at exactly the STPT. That's why you want to CZ pretty much always after finished making your marks and after switching to the mark (STPT) you wish to use. No need to CZ between marks.
And as I understand it, this is with all sensors and is system-wide. You can use CZ from any page where it's available (at least FCR and TGP) and it removes the cursor delta.
And a helpful tip: On MARK page enter 0 and it will set current markpoint as the selected STPT. No need to go to STPT page to switch STPT after creating a markpoint.
 

Hi,
If Cougar DGFT/MSL 3-way switch works the same way as WH boat switch, you don't bind center position. Use the special binds "DOGFIGHT/CENTER" and "MISSILE OVERRIDE/CENTER". Basically these return the control to the center position when you no longer "press" the bound button (i.e. when you return the 3-way switch to center from DGFT or MSL positions).
Here's my binds for TM Warthog throttle. I suppose Cougar has a similar 3-way switch, but button numbering is likely different:

Hope this helps.